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What types of mathematical "sameness" do we have?

From what I've gathered (through studies / this afternoon):

  • Equality between numbers or sets (meaning $A\subset B \;\&\; B\subset A$)
  • Homeomorphism between topological spaces (preserving topological properties)
  • Homomorphism between two algebraic structures, preserving their respective properties
  • Isomorphism in algebra, a bijective homomorphism (related question)
  • Graph isomorphism (added separately because of its importance in graph theory, unlike in abstract algebra)
  • Congruence in algebra (e.g. $a \equiv b \; (\text{mod } n)$)
  • Same cardinality of sets when there exists a bijection between them
  • Congruence (geometry) if one object can be translated into another through an isometry

I am aware that this question might be vague, ill-defined, or too broad since I am asking about objects which I do not yet perhaps know. There might be too many, it might not be clear whether two types are different in the light of this question. I'd simply like to learn more and clarify my existing knowledge. (Also, Math.SE community often gives excellent and eye-opening answers to sometimes even very confused questions - as long as the confusion does not only stem from lack of effort)

Thank you.

Note: I chose the odd title deliberately as to not to hint towards some more mathematical meaning (e.g. equality).

Community
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Dahn
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    Back in February 2015, John Baez did four blog posts on the n-Category Cafe regarding "concepts of sameness." They're a mixture of math and philosophy, so you may find it relevant. – Semiclassical Aug 06 '15 at 13:45
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    To me any relation that is an equivalence relation could be considered "mathematical sameness." There are lots of equivalence relations. Or do you mean something more specific than an equivalence relation? – m_squared Aug 06 '15 at 13:47
  • By the way, a mildly interesting fact is that any set $S$ defines an equivalence relation $\equiv_S$, where:$$a\equiv_Sb\iff\(a\in S\land b\in S)\lor(a\not\in S\land b\not\in S)$$The statement $a\not\equiv_SS$ is equivalent to $a\in S$. Thus, one can replace set theory with "equivalence relation theory," if one wanted to. I don't know of any advantages of thinking of everything as an equivalence relation, though. – Akiva Weinberger Aug 06 '15 at 19:23
  • @Semiclassical thank you, I was so sure something like this had to already exist! – Dahn Aug 07 '15 at 00:29
  • You are absolutely correct, @m_squared but making sense of (the most notable) examples, or of some kind of hierarchies can be very useful. I tried my best to make this less of a "give me examples of mathematics" question and more a "let's find the pattern that somehow unifies and categorizes the notion of sameness" question. How much I succeeded, of course, is a different story. – Dahn Aug 07 '15 at 09:12

3 Answers3

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Geometry has lots of notions of sameness.

Euclidean geometry says two things are the "same" if they are congruent.

Affine geometry says two things are the same if there is an affine transformation between them. (An affine transformation is the sort of "shearing" operation that converts a rectangle to a parallelogram, or a dilation, or an isometry, or a combination of those.) All triangles are affine-congruent. All ellipses are affine-congruent. Lengths are not preserved by affine transformations, but midpoints and collinearity are.

Projective geometry says two things are the same if there is a projective transformation between them. All conic sections are projective-congruent. All quadrilaterals are projective-congruent, IIRC.

This is basically the Kleinian view of geometry, which I learned from this textbook.

Akiva Weinberger
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The notion of computational indistinguishability, whereby two objects are considered "the same" if no efficient (i.e., polynomial-time) algorithm can tell them apart, is central to modern cryptography. (The formal definition involves families of random variables and is a bit messy, this is something most if not all beginners in cryptography struggle with.)

In general, being able to distiguish the two objects under consideration results in "breaking" the proposed cryptographic system, but it is unreasonable to require that this be absolutely impossible because it would result in the impossibility of constructing a practical system satisfying the desired properties (or even of constructing such a system at all). Hence, by merely requiring them to be computationally indistinguishable (as opposed to identical), we accept that it could be possible for an adversary to break the system as long as the computational cost of doing so is (believed to be) sufficiently high to be out of reach of any physical entity. On the other hand, it allows for systems sufficiently practical to be used in practice, as we know them today.

fkraiem
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This could be quite an open-ended question. On the other hand, in the sense that "sameness" can often be expressed as an equivalence relation, we might consider various different kinds of "sameness" the "same".

But perhaps we can just each contribute one or two examples to the list and call that an "answer"?

There are at least two versions of the integral and differential calculus: standard analysis and non-standard analysis. An application of the transfer principle relates these two kinds of analysis, showing that they are in some sense "the same" despite their obvious (and formerly very controversial) differences.

David K
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  • Related: Conservative extensions of theories are "the same" because, by definition, anything that can be proven in the extension (written in the original theory's language) can be proven in the original theory, though the proof in the extension might be shorter. (This is related, because IST — one of the formalisms of nonstandard analysis — is a conservative extension of ZFC. Though I should mention that Robinson's version was entirely within ZFC.) – Akiva Weinberger Aug 06 '15 at 18:09
  • @columbus8myhw This sounds like another answer to the question (at least insofar as one can "answer" this question). – David K Aug 06 '15 at 19:18